def testWeilPairing(self):
# this example was refered to Washington.
e = elliptic.EC([0, 2], 7)
P = [5, 1]
Q = [0, 3]
R = e.WeilPairing(3, P, Q)
self.assertEqual(finitefield.FinitePrimeFieldElement(2, 7), R)
# test case of extension field, characteristic 7
p = 7
r = 11
F = finitefield.FinitePrimeField(p)
PX = uniutil.polynomial({0:3,1:3,2:2,3:1,4:4,5:1,6:1,10:1},F)
Fx = finitefield.FiniteExtendedField(p,PX)
E = elliptic.EC([F.one,-F.one],F)
Ex = elliptic.EC([Fx.one,-Fx.one],Fx)
P = [3,6]
assert E.whetherOn(P)
assert Ex.whetherOn(P)
assert E.mul(11,P) == E.infpoint
Qxcoord = Fx.createElement(6*7**9+7**8+7**6+6*7**3+6*7**2+7+6)
Qycoord = Fx.createElement(3*7**9+6*7**8+4*7**7+2*7**6+5*7**4+5*7**3+7**2+7+3)
Q = [Qxcoord,Qycoord]
assert Ex.whetherOn(Q)
assert Ex.mul(11,Q) == Ex.infpoint
w = Ex.WeilPairing(11, P, Q)
Wp = Fx.createElement(7**9 + 5*7**8 + 4*7**7 + 2*7**5 + 7**4 + 6*7**2)
assert w == Wp
def testWeilPairingIsFunction(self):
# e2 is isomorphic to Z/256 x Z/256
e2 = elliptic.EC([-1, 0], 65537)
P1 = [
finitefield.FinitePrimeFieldElement(30840, 65537),
finitefield.FinitePrimeFieldElement(53250, 65537)
]
self.failIf(256 % e2.pointorder(P1))
P2 = [
finitefield.FinitePrimeFieldElement(10657, 65537),
finitefield.FinitePrimeFieldElement(46245, 65537)
]
self.failIf(256 % e2.pointorder(P2))
weil10 = set(e2.WeilPairing(256, P1, P2) for i in range(10))
# since Weil pairing is a function, the result is always same
self.assertEqual(1, len(weil10))
# Weil pairing is a function E[m]xE[m] -> mu_m
self.assertEqual(e2.basefield.one, weil10.pop()**256)
def testTatePairing_Extend(self):
# this example was refered to Kim Nguyen.
e = elliptic.EC([0, 4], 997)
P = [0, 2]
Q = [747, 776]
R = e.TatePairing_Extend(3, P, P)
W1 = e.TatePairing_Extend(3, P, Q)
W2 = e.TatePairing_Extend(3, Q, P)
self.assertEqual(e.basefield.one, R)
self.assertEqual(finitefield.FinitePrimeFieldElement(304, 997), W1)
self.assertEqual(W1, W2.inverse())
def toFinitePrimeFieldElement(self):
""" get FinitePrimeField(2) element with bijective map.
"""
if self.boolean:
return finitefield.FinitePrimeFieldElement(1, 2)
return finitefield.FinitePrimeFieldElement(0, 2)
